Notes on Relativity and Cosmology - Physics Department, UCSB

66 CHAPTER 3. EINSTEIN AND INERTIAL FRAMES
line marked “friend’s line of simultaneity” below actually represents.
x = 0
f
x =0 us
x =0 us
x = 0
f
t = const
f
friend’s line of
simultaneity
C
Us
α
B
β
t =+1sec
us
t =0
us
t = const
f
t =0
f
A
C
Us
B
In other words, we do not yet understand the rate at which some observer’s clock
ticks in another observer’s reference frame.
To work this out, it is useful to tie time measurements directly to our axioms,
just as we found it useful to tie simultaneity directly to our axioms in section
3.4. That is, we should somehow make a clock out of light! For example, we
can bounce a beam of light back and forth between two mirrors separated by
a known distance. Perhaps we imagine the mirrors being attached to a rod of
fixed length L. Since we know how far apart the mirrors are, we know how long
it takes a pulse of light to travel up and down and we can use this to mark the
passage of time. We have a clock.
3.5.1 Rods in the perpendicular direction
A useful trick is to think about what happens when this ‘light clock’ is held
perpendicular to the direction of relative motion. This direction is simpler
than the direction of relative motion itself. For example, two inertial observers
actually do agree on which events are simultaneous in that direction. To see
this, suppose that I am moving straight toward you (from the front) at some
constant speed. Suppose that you have two firecrackers, one placed one light